Abstract

Abstract—Ca2+ transients measured in failing human ventricular myocytes exhibit reduced amplitude, slowed relaxation, and blunted frequency dependence. In the companion article (O’Rourke B, Kass DA, Tomaselli GF, Kääb S, Tunin R, Marbán E. Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart, I: experimental studies. Circ Res. 1999;84:562–570), O’Rourke et al show that Ca2+ transients recorded in myocytes isolated from canine hearts subjected to the tachycardia pacing protocol exhibit similar responses. Analyses of protein levels in these failing hearts reveal that both SR Ca2+ ATPase and phospholamban are decreased on average by 28% and that Na+/Ca2+ exchanger (NCX) protein is increased on average by 104%. In this article, we present a model of the canine midmyocardial ventricular action potential and Ca2+ transient. The model is used to estimate the degree of functional upregulation and downregulation of NCX and SR Ca2+ ATPase in heart failure using data obtained from 2 different experimental protocols. Model estimates of average SR Ca2+ ATPase functional downregulation obtained using these experimental protocols are 49% and 62%. Model estimates of average NCX functional upregulation range are 38% and 75%. Simulation of voltage-clamp Ca2+ transients indicates that such changes are sufficient to account for the reduced amplitude, altered shape, and slowed relaxation of Ca2+ transients in the failing canine heart. Model analyses also suggest that altered expression of Ca2+ handling proteins plays a significant role in prolongation of action potential duration in failing canine myocytes.

Recent studies using the canine tachycardia pacing-induced model of heart failure12345678 demonstrate that changes in cellular electrophysiological and excitation-contraction (E-C) coupling processes are qualitatively similar to those observed in cells isolated from failing human heart. In human heart failure, IK1 current density measured at hyperpolarized membrane potentials is reduced by ≈50%,910 and density of the transient outward current Ito1 is reduced by ≈75% in subepicardial11 and ≈40% in midmyocardial ventricular cells9 and is unchanged in subendocardial ventricular cells.11 The magnitude of IK1 is reduced by ≈40%, and that of Ito1 by ≈70% in failing canine midmyocardial cells.5 Expression of proteins involved in E-C coupling is also altered in human heart failure. Sarcoplasmic reticulum (SR) Ca2+ ATPase mRNA level,1213141516 protein level,121718 and uptake rate19 are reduced by ≈50% in end-stage heart failure. Na+/Ca2+ exchanger (NCX) mRNA levels are increased by ≈55% to 79%,1220 and NCX protein levels increase 36% to 160%.12202122 Less information is available with regard to NCX function in heart failure. However, Reinecke et al22 reported an 89% increase in sodium-gradient–stimulated 45Ca2+ uptake in human heart sarcolemmal vesicles.

As described in the preceding article by O’Rourke et al,23 alterations of intracellular Ca2+ handling in failing canine midmyocardial ventricular myocytes parallel those observed in human. In particular, the time constant of Ca2+ uptake in the absence of Na+/Ca2+ exchange is prolonged in failing cells (576±83 versus 282±30 ms in controls), suggesting a functional downregulation of the SERCA2a. This observation is consistent with Western blot analyses indicating that SR Ca2+ ATPase protein levels are reduced in failing heart by 28%. Additionally, in the presence of cyclopiazonic acid (CPA, a blocker of the SR Ca2+ ATPase pump), the time constant of Ca2+ extrusion is larger in normal than failing cells (813±269 versus 599±48 ms). This observation is consistent with Western blot analyses indicating a 104% increase in the level of expression of the NCX in failing cells. Taken together, these results suggest that SR Ca2+ uptake is impaired and that Ca2+ extrusion via the NCX is enhanced in myocytes isolated from the failing canine heart in a way that is similar qualitatively to that seen in human patients.

In this article, we use the data of O’Rourke et al23 to develop a computational model of the action potential and of intracellular Ca2+ handling in normal and failing canine ventricular myocytes using biophysically detailed descriptions of both sarcolemmal currents and key components of E-C coupling. With the limits of individual alterations fixed using experimentally derived values, the model is used to quantify the extent to which each parameter (Ito1, IK1, SR Ca2+ ATPase, and NCX) contributes to the overall change in electrical and Ca2+ dynamics in heart failure. The results support the hypothesis that differences in expression of sarcolemmal ion channels and Ca2+ handling proteins measured experimentally are sufficient to account for the altered action potential waveform and Ca2+ transient of the failing canine cardiomyocyte.

Materials and Methods

Normal Canine Ventricular Cell Model

Jafri et al24 have presented a model of Ca2+ handling in the guinea pig ventricular myocyte that incorporates the following: (1) sarcolemmal ion currents of the Luo-Rudy phase II ventricular cell model,25 (2) a state model of the L-type Ca2+ current in which Ca2+-mediated inactivation occurs via the mechanism of mode switching,26 (3) calcium-induced calcium release from SR via ryanodine-sensitive calcium release (RyR) channels using a model adapted from that of Keizer and Levine,27 and (4) a restricted subspace located between the junctional SR (JSR) and T tubules into which both L-type Ca2+ and RyR channels empty. The model of the canine midmyocardial ventricular cell used in this study is derived from this guinea pig ventricular cell model. All dynamic equations, parameters, and initial conditions for this new model are given in the Appendix. The following modifications to the model of Jafri et al24 have been made to better represent properties of canine midmyocardial ventricular cells.

Ito1

Canine epicardial and midmyocardial ventricular cell action potentials exhibit a prominent notch in phase 1 of the action potential that results from the presence of 2 transient outward currents: a Ca2+-independent 4-aminopyridine (4-AP)–sensitive current (Ito1)52829 and a Ca2+-dependent current (Ito2).2930 The Ca2+-independent component Ito1 is modeled on the basis of the formulation of Campbell et al31 for ferret ventricular cells. Peak Ito1 conductance (Gto1) was adjusted to yield a linear plot of peak current density in response to 500-ms–duration voltage-clamp stimuli from a holding potential of –80 mV, with slope 0.3 pA/pF-mV and y-intercept 4.6 pA/pF. This agrees well with experimental measurements reported for canine Ito1 at 37°C by Liu et al28 (see their Figure 10B: slope, 0.28 pA/pF-mV, and y-intercept, 5 pA/pF). Activation rate constants were scaled to yield a time to peak of ≈8 ms at a clamp potential of +10 mV (see Figure 5B⇓ of Tseng and Hoffman).29 Inactivation rate constants were adjusted to yield a decay time constant of ≈20 ms.29 The Ca2+-dependent chloride (Cl–) current Ito2 was not incorporated in this model.

IKr

The delayed rectifier current IK in both canine and guinea pig ventricular myocytes consists of rapid- and slow-activating components known as IKr and IKs, respectively. Models of IKr and IKs in guinea pig ventricular cells have been developed.32 These models have been modified to approximate properties of corresponding currents measured in isolated canine midmyocardial ventricular cells. IKr is described using a closed-open–state model in which forward (K12) and backward (K21) rate constants are exponential functions of voltage (V) with the following form: Parameters of this model are fully constrained by knowledge of the time constant τ(V), defined as at 2 voltages and by knowledge of the steady-state activation function. Activation was fit using a Boltzmann function determined by Liu and Antzelevitch33 (see their Figure 11). The time constant of activation at +5 mV was set to 100 ms,33 and the time constant of deactivation at –60 mV was set to 3000 ms,34 thereby constraining the rate constants K12(V) and K21(V). A fixed increment of 27 ms was added to Equation 2 to bound the time constant away from 0 at depolarized potentials. The maximum conductance G̅Kr was adjusted to yield a tail current density of 0.2 pA/pF in response to a voltage-clamp step to +25 mV for 3.0 seconds, followed by a step to –35 mV for 1.0 seconds, as described by Gintant.35

IKs

The slow-activating delayed rectifier current IKs is present in epicardial, midmyocardial, and endocardial canine ventricular cells. IKs is modeled as described in Zeng et al,32 with the exception that the steady-state activation function is fit using a Boltzmann function determined by Liu and Antzelevitch.33 The voltage-dependent time constant is also shifted by +40 mV in the depolarizing direction to fit the experimental data of Liu and Antzelevitch33 (see their Figure 13). Maximum conductance (G̅Ks) is adjusted to yield a tail current density of 0.4 pA/pF in response to 3.0-second–duration voltage-clamp steps from the holding potential of –35 to +25 mV, followed by a return to the holding potential34 (see Figure 5⇓). The Ca2+ dependence of IKs described in the Luo-Rudy phase II guinea pig model is not included, as there are no experimental data constraining this dependence in canine ventricular cells.

IK1

IK1 is fit using data measured at 22°C in isolated canine midmyocardial ventricular myocytes measured by Kääb et al5 and scaled to 37°C. These data indicate that maximum outward IK1 density is ≈2.5 pA/pF at –60 mV5 (see Reference 5 , Figure 4B⇓). These data also show that IK1 density is nonnegligible at voltages within the plateau range of the canine action potential. For example, IK1 density is 0.3 pA/pF at 0 mV, a value comparable with the density of IKr during the plateau phase of the action potential. The functional representation of IK1 in the Luo-Rudy phase II model can therefore not be used, as it approaches 0 at plateau membrane potentials. An alternative formulation better approximating the canine data is presented in the Appendix.

ICa,L

The model of L-type Ca2+ current used is identical to the mode-switching model presented in Jafri et al,24 with 3 exceptions. First, the voltage dependence of the activation transition rates α(V) and β(V) and the inactivation variable y(V) are shifted by +10 mV in the depolarizing direction to position the peak L-type Ca2+ current in response to voltage-clamp stimuli at +5 mV, as measured experimentally.5 Second, the monotonic decreasing steady-state (voltage-dependent) inactivation function y∞ is modified to have an asymptotic value of 0.2 for large positive membrane potentials V. This modification reproduces the slow component of Ca2+ current observed under voltage-clamp stimuli in canine ventricular cells.537 Finally, peak L-type Ca2+ current density is adjusted to a value of 2.5 pA/pF at a clamp voltage of +5 mV.

Jup

In the model of Jafri et al,24 Ca2+ uptake into network SR (NSR) is modeled using a Hill function with coefficient of 2. Reverse pump rate is assumed to be 0, and Ca2+ leak from NSR to cytoplasm is assumed to be proportional to the gradient of NSR and cytosolic Ca2+ concentrations. Recently, Shannon et al38 have proposed the hypothesis that SR Ca2+ accumulation at rest is not limited by leak of Ca2+ from SR but rather is limited by a reverse component of SR Ca2+ ATPase pump current. They have proposed a new model of the SR Ca2+ ATPase pump that includes forward- and reverse-current components, each with its own binding constant and peak forward and reverse rates (denoted Vmaxf and Vmaxr, respectively).39 The forward mode exhibits slight cooperativity, whereas the reverse mode is noncooperative. The relative magnitudes of forward- and reverse-current components determine whether SR load increases, is constant, or decreases during diastole. The model is presented in the Appendix.

Failing Canine Ventricular Cell Model

Kääb et al5 have shown that in the canine tachycardia pacing-induced model of heart failure, Ito1 and IK1 are downregulated on average by 66% and 32%, respectively, in terminal heart failure. Only the number of expressed channels is changed; the kinetic properties of Ito1 and gating behavior of IK1 are unaltered. On the basis of these data, the effects of terminal heart failure are modeled by reducing the peak conductance of Ito1 and IK1 by the factors indicated above. Downregulation of the SR Ca2+ ATPase is modeled by simultaneous scaling of both the forward and reverse maximum pump rates Vmaxf and Vmaxr by a scale factor, KSR. Upregulation of the NCX is modeled by increasing a scale factor, KNaCa.

Numerical Methods

The dynamical equations in the Appendix are solved on a Silicon Graphics workstation using the Merson modified Runge-Kutta fourth-order adaptive step algorithm (No. 25, Reference 5252 ), with a maximum step size of 100 microseconds and maximum error tolerance of 10–6. The error from all variables is normalized to ensure that each contributes equally to the calculation of global error, as described in Jafri et al.24 Initial conditions listed in the Appendix are used in all calculations, unless noted otherwise. These initial conditions were computed in response to a periodic pulse train of frequency 1 Hz and were determined immediately before the 11th pulse. Action potentials are initiated using 0.1 μAμF–1 current injection for 500 microseconds.

The canine ventricular cell model is used to derive quantitative estimates of the NCX scale factor KNaCa and the SR Ca2+ ATPase scale factor KSR from experimental data by fitting model Ca2+ transient decay rates to those measured experimentally. To do this, a series of 10 voltage-clamp stimuli (–97-mV holding potential, 3-mV step potential, and 200-ms duration) are applied at a frequency of 1 Hz. Ca2+ transient decay rate is estimated from response to the final voltage-clamp stimulus to assure that model SR Ca2+ concentrations have reached equilibrium values.

Results

Figure 1⇓ demonstrates the ability of the model to reconstruct action potentials and Ca2+ transients of both normal and failing canine midmyocardial ventricular myocytes. The solid and dotted lines in Figure 1A⇓ show experimental measurements of normal and failing action potentials, respectively. Model action potentials are shown in Figure 1C⇓. In this figure, the solid line shows a normal action potential. The dashed line shows an action potential when Ito1 is reduced by 66% of the normal values and IK1 by 32% of the normal values (the average percentage reductions observed in terminal heart failure.)5 The dotted line corresponds to these same reductions of Ito1 and IK1, in addition to a 62% reduction of the SR Ca2+ ATPase pump and a 75% increase of the NCX. These values are model-based estimates of the average percentage change in activity of these proteins determined using experimentally derived limits on their function, as described in the following sections.

Model vs experimental action potentials and Ca2+ transients. Each action potential and Ca2+ transient is in response to a 1-Hz pulse train, with responses measured in the steady state. A, Experimentally measured membrane potential as a function of time in normal (solid line) and failing (dotted line) canine myocytes. B, Experimentally measured cytosolic Ca2+ concentration (μmol/L) as a function of time for normal (solid line) and failing (dotted line) canine ventricular myocytes. C, Membrane potential as a function of time simulated using the normal canine myocyte model (solid line), the myocyte model with Ito1 and IK1 downregulation (dashed line; downregulation by 66% and 32%, respectively), and the heart failure model (dotted line; downregulation of Ito1 and IK1 as described previously, KSR=0.38 corresponding to 62% downregulation and KNaCa=0.53 corresponding to 75% upregulation). D, Cytosolic Ca2+ concentration (μmol/L) as a function of time simulated using the normal (solid line) and heart failure (dotted line) model, with parameters as described in panel A. E, L-type Ca2+ current as a function of time for the normal (solid line) and failing (dotted line) cell models. F, Na+/Ca2+ exchange current as a function of time for the normal (solid line) and failing (dotted line) cell models.

The model data of Figure 1C⇑ show that downregulation of Ito1 and IK1 reduces the depth of the phase 1 notch. However, notch depth is larger in the experimental measurements from the failing myocyte (Figure 1A⇑, dotted line) than is predicted by the model (Figure 1C⇑, dashed line). This greater notch depth is due to the presence of the Ca2+-dependent transient outward current Ito2, which is not included in the model. The most significant change in model action potential duration (APD) occurs with upregulation of the NCX and downregulation of the SR Ca2+ ATPase (Figure 1C⇑, dotted line). These 2 changes alone increase APDs at 90% repolarization (APD90) by ≈200 ms.

Figure 1D⇑ illustrates model normal (solid line) and failing (dotted line) Ca2+ transients. Amplitude of the Ca2+ transient is reduced significantly in the heart failure model. Ca2+ transient shape is flattened, duration is prolonged, and relaxation is slowed. These changes are similar qualitatively to those seen in the experimental data of Figure 1B⇑.

Figures 1E⇑ and 1F⇑ show L-type Ca2+ and Na+/Ca2+ exchange currents for normal (solid lines) and failing (dotted lines) model cells. The reduction in peak magnitude of the L-type Ca2+ current seen in Figure 1E⇑ for the failing model cell results from downregulation of Ito1, which reduces depth of the phase 1 notch and therefore driving force during onset of the L-type Ca2+ current. Figure 1E⇑ also shows that L-type Ca2+ current is increased during the later plateau phase of the action potential in failing model cells. The mechanism of this increase will be considered in subsequent sections. Figure 1F⇑ shows that Na+/Ca2+ exchange operates in reverse mode, generating a net outward current during most of the plateau phase of the action potential. The magnitude of this outward current decreases during the plateau phase, and in the failing cell model the current becomes significantly smaller than the inward L-type Ca2+ current.

These simulations demonstrate the ability of the model to reproduce both normal and failing canine myocyte action potentials and Ca2+ transients. The following sections describe application of the model to estimation of the degree of functional change in the NCX and SR Ca2+ ATPase in control and failing myocytes. The approach is as follows: (1) the time constant of Ca2+ decay (τCa) measured with SR function blocked using CPA data is used to estimate the model Na+/Ca2+ exchange scale factor KNaCa; (2) with KNaCa fixed at this value, the model SR Ca2+ ATPase scale factor KSR required to reproduce the τCa measured in physiological solutions is determined; (3) the SR Ca2+ ATPase reduction in heart failure is cross-checked independently by determining the model SR Ca2+ ATPase scale factor required to reproduce the τCa measured under Na+-free conditions (0-Na data) with the model Na+/Ca2+ exchange set to 0; and (4) the model Na+/Ca2+ exchange scale factor is estimated independently from τCa in physiological solutions using the estimate of SR function determined in step 3.

In the preceding article by O’Rourke et al, Ca2+ transients in response to voltage-clamp stimuli were measured in the presence and absence of CPA, a blocker of the SR Ca2+ ATPase pump. In the presence of CPA, Ca2+ transient decay rate (τCa) following termination of a depolarizing voltage step reflects the rate of extrusion of Ca2+ from the cytosol by the NCX (extrusion by the sarcolemmal Ca2+ ATPase is small). Estimates of the NCX pump current scale factor KNaCa may therefore be obtained by setting the model value of KSR to 0 and varying KNaCa until model Ca2+ transient decay rates match those measured experimentally in the presence of CPA. KNaCa may then be fixed at this value and KSR varied until model Ca2+ transient decay rate matches that measured experimentally using physiological solutions. This procedure can be applied to data obtained from both normal and failing cells to assess the extent of functional upregulation and downregulation of the NCX and SR Ca2+ ATPase in heart failure.

To estimate KNaCa, model KSR was set to 0, 10 voltage-clamp steps (holding potential –97 mV, step potential 3 mV, and duration 200 ms) were applied at a frequency of 1 Hz to assure that Ca2+ levels in each model Ca2+ pool were equilibrated, and model τCa was measured by fitting an exponential function to the decay phase of the final Ca2+ transient. Figure 2A⇓ plots model τCa (ordinate, ms) as a function of KNaCa (abscissa) with KSR=0.0 (open triangles). KNaCa=0.30 yields a τCa equal to the average value measured experimentally in normal myocytes in the presence of CPA (813±269 ms). One SD of experimental variability is accounted for by KNaCa values in the interval (0.21, 0.48). This same curve shows that KNaCa=0.53 produces a τCa matching that measured in failing myocytes in the presence of CPA (599±48 ms). One SD experimental variability is encompassed by KNaCa values in the interval (0.48, 0.60). Assuming the normal value of KNaCa to be 0.30, these data suggest a functional upregulation of the NCX in heart failure in the range of 60% to 100%, with average value ≈75%.

Figure 2B⇑ plots model τCa (ordinate, ms) as a function of KSR (abscissa). The curve marked with open circles plots this dependence when KNaCa is constant at the normal value estimated above (KNaCa=0.30). The experimental value of τCa measured in normal myocytes using physiological solutions is 219±36 ms. The maximum forward and reverse SR Ca2+ ATPase pump rates Vmaxf and Vmaxr given in Table 4 of the Appendix have been selected to yield a similar time constant when KSR=1.0. Measured variation about this value is accounted for by KSR values in the interval (0.85, 1.15).

The experimental value of τCa measured in failing myocytes using physiological solutions is 292±23 ms. Dependence of model τCa on KSR when KNaCa is fixed at the value estimated for failing canine myocytes (0.53) is shown by the curve labeled with open squares in Figure 2B⇑. KSR=0.38 yields a model τCa equal to that observed experimentally. The experimental deviation of τCa is accounted for by KSR values in the interval (0.26, 0.51). Assuming the average value of KSR in normal cells to be 1.0, these data suggest a functional downregulation of the SR Ca2+ ATPase pump in heart failure in the range of 49% to 74%, with average value 62%.

To provide a second, independent measure of altered Ca2+ handling protein expression in heart failure, O’Rourke et al23 have measured τCa in the presence and absence of Na+/Ca2+ exchange by removing Na+ ions from both intracellular and extracellular solutions. In the absence of Na+/Ca2+ exchange, τCa reflects primarily the rate of Ca2+ uptake from the cytosol by the SR Ca2+ ATPase pump. Estimates of the SR Ca2+ ATPase pump rate scale factor KSR under 0-Na conditions may therefore be obtained by setting the model value of KNaCa to 0 and varying KSR until simulated voltage-clamp Ca2+ transient decay rates match those measured experimentally. Once the model value of KSR is constrained, KNaCa can then be determined by changing its value until model Ca2+ transient decay rates match those measured experimentally using physiological solutions. This procedure can be applied to data obtained from both normal and failing cells to assess the extent of functional upregulation and downregulation of the NCX and SR Ca2+ ATPase in heart failure.

To mimic 0-Na conditions, KNaCa was set equal to 0. KSR was then varied, and the time constant for Ca2+ reuptake into SR was computed. Model τCa values are plotted as a function of KSR in Figure 2B⇑ (open triangles). Experimentally measured values of this time constant are 282±30 ms in normal and 576±83 ms in failing canine ventricular cells studied under 0-Na conditions. A KSR value of 1.0 accounts for τCa measured experimentally in normal cells (282±30 ms), and values in the interval (0.92, 1.07) account for the observed SD in these measurements. This estimate of the average KSR value in normal myocytes based on block of the NCX agrees with that estimated using the CPA data. A KSR value of 0.51 accounts for the average τCa measured experimentally in failing cells (576±83 ms), and KSR values in the interval (0.46, 0.59) account for the SD. Assuming the normal KSR value to be 1.0, these data suggest a functional downregulation of the SR Ca2+ ATPase pump in failing myocytes in the range of 41% to 54%, with average value 49%. This estimate of SR Ca2+ ATPase downregulation is qualitatively similar to that obtained using CPA.

Dependence of model τCa on KNaCa when KSR is fixed at the value estimated for normal canine myocytes (1.0) is shown by the curve labeled with open circles in Figure 2A⇑. KNaCa=0.22 yields a model τCa equal to that observed experimentally using physiological solutions (219±36 ms). Experimental deviation of τCa is accounted for by KNaCa values in the interval (0.13, 0.43). Dependence of model τCa on KNaCa when KSR is fixed at the value estimated for failing canine myocytes under 0-Na conditions (0.51) is shown by the curve labeled with open squares in Figure 2A⇑. KNaCa=0.35 yields a model τCa equal to the average value observed experimentally using physiological solutions (292±23 ms). Experimental deviation of τCa is accounted for by KNaCa values in the interval (0.26, 0.46). Assuming the normal value of KNaCa to be 0.22, these data suggest a functional upregulation of the NCX in heart failure in the range of 18% to 109%, with average value 38%. This estimate of altered expression of NCX in heart failure has greater variability than that obtained previously using the CPA data but is consistent in that it also indicates increased expression.

The above analyses provide estimates of KSR and KNaCa in normal and failing myocytes. Results indicate functional downregulation of the SR Ca2+ ATPase pump and upregulation of the NCX in heart failure. The parametric dependence of model cytosolic Ca2+ transients on KSR and KNaCa is examined next.

Model cytosolic Ca2+ concentration (ordinate, μmol/L) versus time (abscissa, seconds) is shown in Figure 3A⇓ as KSR is varied. In these simulations, KNaCa is constant at the value estimated using CPA data from normal cells (KNaCa=0.30). KSR is varied from 1.0 to 0.0 in steps of 0.1. Ca2+ transients are in response to a 1-Hz voltage-clamp stimulus (holding potential –97 mV, step potential 3 mV, and duration 200 ms). Response to the final stimulus of 10 stimulus cycles is shown, with the time origin translated to 0 seconds. These data show that reduction of the model SR Ca2+ ATPase pump, simulating the effects of downregulation of this pump in heart failure, reduces the amplitude of the early peak of the Ca2+ transient (marked by the arrow). This early peak disappears as KSR approaches 0. Figure 3B⇓ shows JSR Ca2+ levels for each of the responses in Figure 3A⇓. Reduction of the early peak in the data of Figure 3A⇓ coincides with depletion of JSR Ca2+ at small values of KSR. Thus, the early peak in the model Ca2+ transient is generated by Ca2+ release from JSR, and the slow second peak, which is present even when JSR is depleted, results from influx of Ca2+ through sarcolemmal L-type Ca2+ channels and reverse-mode Na+/Ca2+ exchange. As KSR decreases, Ca2+ levels in JSR decrease, and the Ca2+ transient becomes reduced in peak amplitude. The Ca2+ transient exhibits a decrease, no change, or an increase of amplitude during the course of the voltage-clamp stimulus, depending on the value of KSR. Decay rate of the Ca2+ transient decreases with decreasing KSR values, as shown in the data of Figure 3A⇓, as well as Figure 2B⇑ (open circles).

A, Cytosolic Ca2+ concentration (μmol/L) as a function of time in response to a 10-second–duration periodic sequence of voltage-clamp stimuli with frequency of 1 Hz. Holding potential is –97 mV, and clamp potential is +3 mV, with duration of 200 ms. Only response to the final stimulus is shown. A family of responses is shown in which KNaCa is constant at 0.30 (value estimated in CPA experiments with normal canine myocytes), and KSR is varied from 1.0 to 0.0 in steps of 0.1. B, Model JSR Ca2+ concentration in response to the stimuli in panel A.

Figure 4A⇓ shows model cytosolic Ca2+ concentration (ordinate, μmol/L) versus time (abscissa, seconds) as KNaCa is varied in steps of 0.5 from 0.5 to 2.5. A plot for KNaCa=0.25 is also shown. KSR is constant at the value estimated using CPA data from normal myocytes (KSR=1.0). Voltage clamp steps from –97 mV to +3 mV with 200 ms duration are applied at a rate of 1.0 Hz. The final Ca2+ transient in a sequence of 10 is displayed, with the time origin translated to 0 seconds. There are 3 effects of increased KNaCa. These are (1) increased rate of Ca2+ extrusion and lower diastolic Ca2+ at the holding potential, (2) reduction in Ca2+ transient amplitude in response to the +3 mV voltage-clamp step, and (3) “flattening” of the Ca2+ transient during the voltage-clamp step. The increased Ca2+ extrusion at the holding potential is a direct consequence of increased NCX activity when the exchanger is operating in the forward mode at the –97-mV holding potential, as shown in Figure 4B⇓. This figure also shows that the NCX operates in reverse mode at the +3 mV clamp potential, thus generating Ca2+ influx. The reduction in Ca2+ transient amplitude in response to the voltage step is a consequence of the fact that total Ca2+ extrusion at the holding potential is greater than total Ca2+ influx at the step potential. This produces a smaller Ca2+ transient through reductions in SR Ca2+ loading and therefore a smaller Ca2+ release. The flattening of the Ca2+ transient with increased KNaCa is a direct consequence of increased Ca2+ influx during the voltage step, as shown in Figure 4B⇓. Decreased KNaCa values also produce smaller Ca2+ transient decay rates, as seen by the data of Figure 4A⇓, as well as Figure 2A⇑ (open circles).

A, Cytosolic Ca2+ concentration (μmol/L) as a function of time in response to a 10-second–duration periodic sequence of voltage-clamp stimuli with frequency of 1 Hz. Holding potential is –97 mV, and clamp potential is +3 mV, with duration of 200 ms. Only response to the final stimulus is shown. A family of responses is shown in which KNaCa is varied from 0.5 to 2.5 in steps of 0.5 (response for KNaCa=0.25 is also shown), while KSR is held constant at the value accounting for the Ca2+ relaxation rate measured in normal myocytes under 0-Na conditions. B, Model NaCa exchange current INaCa as a function of time in response to the stimuli described in Figure 3A⇑.

Figure 5A⇓ shows model Ca2+ transients in response to a 1-Hz voltage-clamp pulse train. These transients were computed using KSR and KNaCa parameter values determined from the experimental series in the presence and absence of CPA. The solid line is the normal model Ca2+ transient (KNaCa=0.30 and KSR=1.0). The peak Ca2+ level (480 nmol/L) agrees well with the value measured experimentally in normal myocytes (450±75 nmol/L).23 The dotted line is the model Ca2+ transient computed using the average KNaCa (0.53) and KSR (0.38) values for failing myocytes. The remaining 2 Ca2+ transients (dashed lines) correspond to KNaca and KSR values selected at ±1 SD from the average for failing myocytes. The short dashed line represents parameter choices producing a high degree of SR unloading (large NCX activity, KNaCa=0.60; small SR Ca2+ ATPase activity, KSR=0.26). The long-dashed line represents parameter choices that minimize SR unloading (small NCX activity, KNaCa=0.48; large SR Ca2+ ATPase activity, KSR=0.51). These data show that as KNaCa is increased from a normal value of 0.30 (taking on values of 0.48, 0.53, and 0.60) and KSR is decreased from the normal value of 1.0 (taking on values 0.51, 0.38, and 0.26), Ca2+ transient peak decreases monotonically from the normal value of 480 nmol/L, taking on values of 300, 266, and 230 nmol/L. These values agree well with the average experimental values measured in failing cells of 230±40 nmol/L.23

A, Cytosolic Ca2+ concentration (μmol/L) as a function of time in response to a 10-second–duration periodic sequence of voltage-clamp stimuli with frequency of 1 Hz. Holding potential is –97 mV, and clamp potential is +3 mV, with duration of 200 ms. Only response to the final stimulus is shown. Results of 4 simulations are shown. Solid line indicates normal model Ca2+ transient (KNaCa=0.30; KSR=1.0). Dotted line indicates the average failing model Ca2+ transient computed using values of KSR and KNaCa estimated in the presence/absence of CPA, as described in the text (KSR=0.38; KNaCa=0.53). Short-dashed lines show failing model Ca2+ transients computed using KSR=0.26 and KNaCa=0.60. Long-dashed lines show failing model Ca2+ transients computed using KSR=0.51 and KNaCa=0.48. B, Experimentally measured cytosolic Ca2+ concentration (μmol/L) vs time in response to voltage-clamp stimuli (as described in panel A) in normal (solid line) and failing (dotted line) canine ventricular myocytes. C, L-type Ca2+ current as a function of time for the stimuli and model parameters described in panel A. D, Subspace Ca2+ concentration (μmol/L) as a function of time for the stimuli and model parameters described in panel A.

Figure 5B⇑ shows a Ca2+ transient measured experimentally. The amplitude and waveform of the model predictions in Figure 5A⇑ are in close agreement with these experimental data.

Figure 5C⇑ shows a plot of the L-type Ca2+ current during the Ca2+ transients of Figure 5A⇑. The parameter changes have relatively little effect on peak current, but increases in KNaCa or decreases in KSR produce a monotonic increase in the late component of the L-type Ca2+ current. As shown in Figure 5D⇑, these same parameter changes also produce monotonic decreases of the subspace Ca2+ transient peak. Thus, the increase in the late component of L-type Ca2+ current seen in Figure 5C⇑ results from a decrease in Ca2+-mediated inactivation of this current due to reductions in magnitude of the subspace Ca2+ transient, which is in turn a consequence of reduced SR Ca2+ load. As can be appreciated by examining the magnitude of the change in L-type Ca2+ current density with alterations in Ca2+ handling, this late component of the L-type Ca2+ current would be expected to play an important role in determining the action potential plateau. This suggests that in heart failure, alterations in the expression of Ca2+ handling proteins that decrease SR Ca2+ load and reduce the amplitude of the Ca2+ transient may contribute substantially to prolongation of APD by reducing Ca2+-mediated inactivation of the L-type Ca2+ current.

Discussion

In this article, we present a model of the canine midmyocardial action potential and Ca2+ transient. The model is used to estimate the magnitude of SR Ca2+ ATPase pump rates and NCX current in normal and failing myocytes23 using 2 methods. In the first method, model SR Ca2+ ATPase current is set to 0, and the NCX current is scaled to yield Ca2+ relaxation time constants in response to voltage-clamp stimuli matching those measured experimentally in normal and failing myocytes in the presence of CPA, a blocker of the SR Ca2+ ATPase. The extent of functional upregulation of the NCX in heart failure estimated using this approach is in the range of 60% to 100%, with average value 75%. Having constrained the model NCX current, model SR Ca2+ ATPase pump current is then estimated by matching the model Ca2+ relaxation rate to experimental data obtained in the absence of CPA. Comparison of model SR Ca2+ ATPase pump currents estimated for normal and failing myocytes suggests a functional downregulation in heart failure in the range of 49% to 74%, with average value 62%.

In the second method, model NCX current is set to 0, and the SR Ca2+ ATPase current is scaled to yield Ca2+ relaxation time constants matching those measured experimentally under 0-Na conditions. Functional downregulation of the SR Ca2+ ATPase current in heart failure estimated using this approach is in the range of 41% to 54%, with average value 49%. Having constrained the model SR Ca2+ ATPase current, NCX current is estimated by matching the model Ca2+ relaxation rate to experimental data obtained in control intracellular and extracellular sodium concentrations. Functional NCX upregulation in heart failure estimated using this approach is in the range of 18% to 109%, with average value 38%.

Analysis of protein levels in canine hearts subjected to the tachycardia pacing protocol reveal that both SR Ca2+ ATPase and phospholamban proteins are reduced on average by 28%23 and that NCX protein is increased on average by 104%.23 Both steady-state mRNA and expressed levels of E-C coupling proteins in failing human ventricular cells have been measured. The majority of reports agree that there is a ≈50% reduction of: (1) mRNA encoding the SR Ca2+ ATPase pump,1213141516 (2) expressed SR Ca2+ ATPase protein level,121718 and (3) direct SR Ca2+ ATPase uptake rate during heart failure.19 There is a 55% to 79% increase in Na-Ca exchanger mRNA levels,1220 a 36% to 160% increase in expressed protein levels,12202122 and an approximate factor of 2 increase in Na+/Ca2+ exchange activity in human heart failure.22

The model-based estimates of functional upregulation and downregulation of the NCX and SR Ca2+ ATPase pump reported here are consistent with these reports. Model estimates of average SR Ca2+ ATPase functional downregulation are 49% and 62%, depending on the estimation methods used. These values agree well with estimates of mRNA level, protein level, and SR Ca2+ ATPase uptake rate measured in human heart failure, but suggest a slightly larger degree of downregulation than indicated by measurements of protein level in canine tachycardia pacing-induced heart failure23 (28%). Model estimates of average NCX upregulation are 38% and 75%. These estimates agree well with measured increases in mRNA levels in human heart failure and are within the range of variability of measured NCX protein levels in human heart failure. However, the model estimates are slightly lower than is suggested by the increased protein levels measured in the failing canine heart.23

Ca2+ transients measured in failing human and canine ventricular myocytes exhibit reduced amplitude and slowed relaxation.540414243 Model simulations of Ca2+ transients in response to voltage-clamp stimuli reported here demonstrate that the altered expression of the NCX and SR Ca2+ ATPase pump measured in failing canine myocytes is sufficient to account for these properties. Both changes contribute to reduced SR Ca2+ load and release and therefore reduced amplitude of the early Ca2+ transient peak (Figures 3A⇑ and 4A⇑). The shape of the Ca2+ transient is also controlled by both NCX and SR Ca2+ ATPase levels. As the Ca2+ ATPase pump is downregulated (Figure 3A⇑), the shape of the plateau portion of the voltage-clamp Ca2+ transient changes from negative to 0, then to positive slope. This change in slope is produced by a decrease in early Ca2+ release from JSR, which in turn increases the dependence of Ca2+ transient shape on Ca2+ entry through the L-type Ca2+ channel. Upregulation of NCX also influences Ca2+ transient shape, tending to flatten the Ca2+ transient plateau by increasing reverse-mode Ca2+ entry at depolarized potentials (Figure 4A⇑). The interplay between both of these factors accounts for the flattened Ca2+ transient shape seen in failing myocytes (Figure 1D⇑, model; Figure 1B⇑, experimental data).

Model Ca2+ transients in response to voltage-clamp stimuli exhibit a “knob” at the early peak of the transient (see Figure 3A⇑, for example) that does not appear to be present in the experimental data. This knob disappears as the SR Ca2+ level becomes small (Figure 3A⇑), indicating that the knob is dependent on SR Ca2+ release. The knob is likely an artifact of model construction. All SR Ca2+ release in this model occurs from a single functional unit, defined as a set of L-type Ca2+ channels, RyR channels, and the subspace within which they interact. Stern has referred to such models as common pool models.44 The knob reflects a large, single Ca2+ release event from this single functional unit. In contrast, real cardiac cells have a large number of functional units in which there is local control of calcium-induced calcium release. We have recently implemented a local control model of Ca2+ release consisting of an ensemble of functional units, in which each functional unit is defined as an L-type Ca2+ channel interacting with a small set of RyR channels through a diadic space. Both L-type Ca2+ channels and RyR channels are modeled stochastically using the channel models presented in Jafri et al.24 In such a model, the stochastic nature of RyR channel openings produces a variable latency of Ca2+ release in each functional unit. The Ca2+ transients computed using this model exhibit the property of gradedness and do not exhibit the knob seen in Figure 3A⇑ due to temporal smearing of Ca2+ release times.

A recent study has put forth the hypothesis that coupling between L-type Ca2+ channels and RyR channels may be altered in heart failure and that this altered coupling leads to a reduction in amplitude of the Ca2+ transient.45 The results presented here cannot refute this hypothesis. Indeed, structurally detailed models of RyR channel and L-type Ca2+ channel interactions in the diadic space predict a strong dependence of these interactions on geometric factors.464748 However, the results reported here indicate that such an assumption is not necessary to account for reduced amplitude of Ca2+ transients in failing myocytes. Rather, these simulations indicate that the altered expression of Ca2+ handling proteins reported by several different groups in both failing human and canine myocytes could account for changes in Ca2+ transient amplitude and shape.

The data of Figure 1⇑ demonstrate that downregulation of the outward repolarizing currents IK1 and Ito1, together with altered expression of the NCX and SR Ca2+ ATPase pump, can account for differences in both action potential and Ca2+ transient shape in heart failure. However, the data of Figure 1C⇑ also indicate that downregulation of IK1 and Ito1, at least to the extent measured on average in failing cells, has a small effect on APD. Instead, altered expression of Ca2+ handling proteins plays a significant role in APD prolongation.

It is not surprising that downregulation of model IK1 has only a modest impact on APD, as IK1 is primarily responsible for the terminal phase of repolarization. However, the finding that reduction of model Ito1 has only a small effect on APD differs from the experimental results of Kääb et al5 in dog myocytes and of Beuckelmann et al9 in human cells. These experiments were performed using EGTA as an intracellular Ca2+ buffer. This buffering minimizes the modulatory effects of Ca2+ and thus enhances the relative influence of outward K currents on action potential characteristics. When effects of EGTA buffering are simulated in the model described in this article, block of Ito1 has a greater influence on APD. An example is shown in Figure 6⇓. The Ca2+ buffering effects of EGTA were modeled using the fast buffering approximation developed by Wagner and Keizer,49 with EGTA= 10 mmol/L and the dissociation constant Km=0.15 μmol/L. Block of Ito1 by 95% increases APD90 by 73 ms, or ≈25% of the control value. These results again emphasize the important modulatory role of Ca2+ on action potential characteristics in the canine myocyte.

Membrane potential as a function of time for the normal model (solid line) and for the model with a 95% reduction in magnitude of Ito1 (dotted line). Simulations are done in the presence of EGTA (10 mmol/L; Km=0.15 μmol/L), using the fast buffering approximation of Wagner and Keizer.49 Response to fifth stimulus at a cycle length of 1200 ms is shown.

It is also possible that 4-AP block of K currents other than Ito1 occurred in the Kääb et al5 experiments, but that such effects were not resolvable. Steady-state current-voltage relations were measured in the presence and absence of 4-AP to assess whether or not this was the case. Data are shown in Figure⇑ 10C of Kääb et al5 and indicate that experimental variability in steady-state current at 0 mV (a potential near that of the action potential plateau) is roughly ±1.0 pA/pF. The sum of model outward currents Ito1, IK1, IKr, and IKs during the plateau is comparable with the magnitude of this variability in the experimental measurements (≈1.0 pA/pF). Genetic approaches for selective suppression of Ito150,51 may turn out to be more useful than pharmacological approaches in determining the influence of this current on APD.

The model predicts that one important mechanism of APD prolongation in heart failure is that shown in Figures 1⇑ and 5⇑. Under conditions of reduced SR Ca2+ release, there is less Ca2+-mediated inactivation of the L-type Ca2+ current. The resulting increase of inward current, as shown for voltage-clamp stimuli in Figure 5C⇑ and for action potentials in Figure 1E⇑, helps to maintain and prolong the plateau phase of the action potential. Investigation into the relative contribution of the various Ca2+-regulatory mechanisms and Ca2+-dependent membrane currents in determining the action potential shape and duration is an important area for future experimental and modeling studies.

Membrane Potential

Ca2+ Handling Mechanisms

Acknowledgments

This research was supported by National Science Foundation grant BIR91-17874, National Institutes of Health grant HL60133, NIH Specialized Center of Research on Sudden Cardiac Death grant P50 HL52307, Silicon Graphics Inc, and the Whitaker Foundation.

Footnotes

This manuscript was sent to Harry A. Fozzard, Consulting Editor, for review by expert referees, editorial decision, and final disposition.

Nabauer M, Beuckelmann DJ, Uberfuhr P, Steinbeck G. Regional differences in current density and rate-dependent properties of the transient outward current in subepicardial and subendocardial myocytes of human left ventricle. Circulation.1996;93:168–177.